It remains nevertheless an interesting issue to prove that the Cauchy problem for the Burgers equation (REF ) in the Sobolev space \(H^s(\mathbb {R})\)  with \(s<\frac{3}{2}\)  is ill-posed. However, due to the absence of the embedding \(H^s(\mathbb {R})\hookrightarrow B^1_{\infty ,\infty }(\mathbb {R})\)  for \(s<3/2\) , the method in [1]}, [2]} is invalid for \(s<3/2\) .
In this paper, we shall develop a new method to study this problem and give a partial answer.
